Technical Field
This disclosure relates to linear transforms of diffusion MRI data.
Description of Related Art
Transformations of Fourier data may be sampled on the surface of a sphere. E(q) may be a three-dimensional Fourier transform of an unknown real-valued function ƒ(x), with:E(q)=ƒ(x)exp(−i2πqTx)dx.  (1)
E(q) may only be observed at points q from a 2-sphere of radius ρ, i.e., qερ2, where ρ2={qε3: ∥q∥l2=ρ}.
Spherical Fourier sampling can provide an accurate model of a data acquisition procedure in high angular resolution magnetic resonance diffusion imaging (HARDI). A spherical linear transform method known as the Funk-Radon Transform (FRT) has been previously proposed by Tuch et al. in this context, see, Tuch, D. S., 2002, “Diffusion MRI of Complex Tissue Structure”, Ph.D. thesis. Massachusetts Institute of Technology; Tuch, D. S., Reese, T. G., Wiegell, M. R., Wedeen, V. J., 2003, “Diffusion MRI of complex neural architecture”, Neuron 40, 885-895; Tuch, D. S., 2004, “Q-ball imaging”, Magn. Reson. Med. 52, 1358-1372. In particular, the FRT transforms the measured data according toRG{E(q)}(u)=E(q)G(uTq)dq  (2),where G(uTq)=δ(uTq) and δ(•) denotes the Dirac delta function. In this notation, the delta function is being used to define a contour integral around the equator of ρ2 that is perpendicular to u.
The FRT is useful in the context of HARDI data because it allows computation of an approximate diffusion orientation distribution function (ODF), which quantifies information about the amount of diffusion along any given orientation u. This diffusion orientation information can be valuable in the study of anisotropic structures like white matter fibers in the central nervous system (CNS), because the diffusion ODF for a given voxel is often similar in shape to the distribution of white matter fibers within that voxel, see, Seunarine, K. K., Alexander, D. C., 2009, “Multiple fibers: beyond the diffusion tensor, in: Johansen-Berg, H., Behrens, T. E. J. (Eds.), Diffusion MRI: from quantitative measurement to in vivo neuroanatomy”, Academic Press. pp. 55-72; Assemlal, H. E., Tschumperlè, D., Brun, L., Siddiqi, K., 2011, “Recent advances in diffusion MRI modeling: Angular and radial reconstruction”, Med. Image Anal. 15, 369-396. The combination of the FRT with HARDI data is also known as Q-ball imaging (QBI), see, Tuch, D. S., 2004, “Q-ball imaging”, Magn. Reson. Med. 52, 1358-1372.
Many alternatives to the FRT have been proposed in recent years for estimating orientation information from the same kind of HARDI data, including multitensor models, see, Alexander, A. L., Hasan, K. M., Lazar, M., Tsuruda, J. S., Parker, D. L., 2001, “Analysis of partial volume effects in diffusion-tensor MRI”, Magn. Reson. Med. 45, 770-780; Tuch, D. S., Reese, T. G., Wiegell, M. R., Makris, N., Belliveau, J. W., Wedeen, V. J., 2002, “High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity”, Magn. Reson. Med. 48, 577-582; Hosey, T., Williams, G., Ansorge, R., 2005, “Inference of multiple fiber orientations in high angular resolution diffusion imaging”, Magn. Reson. Med. 54, 1480-1489; Kreher, B. W., Schneider, J. F., Mader, I., Martin, E., Hennig, J., Il'yasov, K. A., 2005. “Multitensor approach for analysis and tracking of complex fiber configurations”, Magn. Reson. Med. 54, 1216-1225; Peled, S., Friman, O., Jolesz, F., Westin, C. F., 2006, “Geometrically constrained two-tensor model for crossing tracts in DWI”, Magn. Reson. Imag. 24, 1263-1270; Behrens, T. E. J., Johansen-Berg, H., Jbabdi, S., Rushworth, M. F. S., Woolrich, M. W., 2007, “Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?”, NeuroImage 34, 144-155; Jian, B., Vemuri, B. C., O″zarslan, E., Carney, P. R., Mareci, T. H., 2007, “A novel tensor distribution model for the diffusion-weighted MR signal”, NeuroImage 37, 164-176; Ramirez-Manzanares, A., Rivera, M., Vemuri, B. C., Carney, P., Mareci, T., 2007, “Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry”, IEEE Trans. Med. Imag. 26, 1091-1102; Pasternak, O., Assaf, Y., Intrator, N., Sochen, N., 2008, “Variational multipletensor fitting of fiber-ambiguous diffusion-weighted magnetic resonance imaging voxels”, Magn. Reson. Imag. 26, 1133-1144; Melie-Garcia, L., Canales-Rodriguez, E. J., Aleman-Gomez, Y., Lin, C. P., Iturria-Medina, Y., Valdes-Hernandez, P. A., 2008, “A Bayesian framework to identify principal intravoxel diffusion profiles based on diffusion-weighted MR imaging”, NeuroImage 42, 750-770; Leow, A. D., Zhu, S., Zhan, L., McMahon, K., de Zubicaray, G. I., Meredith, M., Wright, M. J., Toga, A. W., Thompson, P. M., 2009, “The tensor distribution function”, Magn. Reson. Med. 61, 205-214; Tabelow, K., Voss, H. U., Polzehl, J., 2012, “Modeling the orientation distribution function by mixtures of angular central Gaussian distributions”, J. Neurosci. Methods 203, 200-211, higher-order generalizations of tensor models, see Alexander, D. C., Barker, G. J., Arridge, S. R., 2002, “Detection and modeling of non-Gaussian apparent diffusion coeffcient profiles in human brain data”, Magn. Reson. Med. 48, 331-340; Frank, L. R., 2002, “Characterization of anisotropy in high angular resolution diffusion-weighted MRI”, Magn. Reson. Med. 47, 1083-1099; Özarslan, E., Mareci, T. H., 2003, “Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging”, Magn. Reson. Med. 50, 955-965; Liu, C., Bammer, R., Acar, B., Moseley, M. E., 2004, “Characterizing non-Gaussian diffusion by using generalized diffusion tensors”, Magn. Reson. Med. 51, 924-937; Schultz, T., Seidel, H. P., 2008, “Estimating crossing fibers: A tensor decomposition approach”, IEEE Trans. Vis. Comput. Graphics 14, 1635-1646; Barmpoutis, A., Hwang, M. S., Howland, D., Forder, J. R., Vemuri, B. C., 2009, “Regularized positive-definite fourth order tensor field estimation from DWMRI”, NeuroImage 45, S153-S162; Liu, C., Mang, S. C., Moseley, M. E., 2010, “In vivo generalized diffusion tensor imaging (GDTI) using higher-order tensors (HOT)”, Magn. Reson. Med. 63, 243-252; Florack, L., Balmashnova, E., Astola, L., Brunenberg, E., 2010, “A new tensorial framework for single-shell high angular resolution diffusion imaging”, J. Math. Imaging Vis. 38, 171-181, directional function modeling, see Kaden, E., Knösche, T. R., Anwander, A., 2007, “Parametric spherical deconvolution: Inferring anatomical connectivity using diffusion MR imaging”, NeuroImage 37, 474-488; Rathi, Y., Michailovich, O., Shenton, M. E., Bouix, S., 2009, “Directional functions for orientation distribution estimation”, Med. Image Anal. 13, 432-444; Rathi, Y., Malcolm, J. G., Michailovich, O., Westin, C. F., Shenton, M. E., Bouix, S., 2010, “Tensor kernels for simultaneous fiber model estimation and tractography”, Magn. Reson. Med. 64, 138-148, spherical polar Fourier expansion, see Assemlal, H. E., Tschumperlè, D., Brun, L., 2009, “Efficient and robust computation of PDF features from diffusion MR signal”, Med. Image Anal. 13, 715-729; Assemlal, H. E., Tschumperlè, D., Brun, L., Siddiqi, K., 2011, “Recent advances in diffusion MRI modeling: Angular and radial reconstruction”, Med. Image Anal. 15, 369-396, independent component analysis, see Singh, M., Wong, C. W., 2010, “Independent component analysis-based multifiber streamline tractography of the human brain”, Magn. Reson. Med. 64, 1676-1684, sparse spherical ridgelet modeling, see Michailovich, O., Rathi, Y., 2010, “On approximation of orientation distributions by means of spherical ridgelets”, IEEE Trans. Image Process. 19, 461-477, diffusion circular spectrum mapping, see Zhan, W., Stein, E. A., Yang, Y., 2004, “Mapping the orientation of intravoxel crossing fibers based on the phase information of diffusion circular spectrum”, NeuroImage 23, 1358-1369, deconvolution, see Tournier, J. D., Calamante, F., Connelly, A., 2007, “Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution”, NeuroImage 35, 1459-1472; Anderson, A. W., 2005. “Measurement of fiber orientation distributions using high angular resolution diffusion imaging”, Magn. Reson. Med. 54, 1194-1206; Descoteaux, M., Deriche, R., Knösche, T. R., Anwander, A., 2009, “Deterministic and probabilistic tractography based on complex fibre orientation distributions”, IEEE Trans. Med. Imag. 28, 269-286; Patel, V., Shi, Y., Thompson, P. M., Toga, A. W., 2010, “Mesh-based spherical deconvolution: A flexible approach to reconstruction of non-negative fiber orientation distributions”, NeuroImage 51, 1071-1081; Yeh, F. C., Wedeen, V. J., Tseng, W. Y. I., 2011, “Estimation of fiber orientation and spin density distribution by diffusion deconvolution”, NeuroImage 55, 1054-1062; Reisert, M., Kiselev, V. G., 2011, “Fiber continuity: An anisotropic prior for ODF estimation”, IEEE Trans. Med. Imag. 30, 1274-1283, the diffusion orientation transform, see Özarslan, E., Shepherd, T. M., Vemuri, B. C., Blackband, S. J., Mareci, T. H., 2006, “Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT)”, NeuroImage 31, 1086-1103; Canales-Rodrìguez, E. J., Lin, C. P., Iturria-Medina, Y., Yeh, C. H., Cho, K. H., Melie-Garcìa, L., 2010, “Diffusion orientation transform revisited”, NeuroImage 49, 1326-1339, estimation of persistent angular structure, see Jansons, K. M., Alexander, D. C., 2003, “Persistant angular structure: new insights from diffusion magnetic resonance imaging data”, Inverse Probl. 19, 1031-1046, generalized q-sampling imaging, see Yeh, F. C., Wedeen, V. J., Tseng, W. Y. I., 2010, “Generalized q-sampling imaging”, IEEE Trans. Med. Imag. 29, 1626-1635, and orientation estimation with solid angle considerations, see Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009, “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010. “Reconstruction of the orientation distribution function in single-and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2010, “A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform”, NeuroImage 49, 1301-1315. However, the FRT has a unique combination of useful characteristics: it does not require a strict parametric model of the diffusion signal, it is linear and its theoretical characteristics can be explored analytically, and it can be computed very quickly using efficient algorithms, see Anderson, A. W., 2005, “Measurement of fiber orientation distributions using high angular resolution diffusion imaging”, Magn. Reson. Med. 54, 1194-1206; Hess, C. P., Mukherjee, P., Han, E. T., Xu, D., Vigneron, D. B., 2006, “Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis”, Magn. Reson. Med. 56, 104-117; Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R., 2007, “Regularized, fast, and robust analytical Q-ball imaging”, Magn. Reson. Med. 58, 497-510; Kaden, E., Kruggel, F., 2011, “A reproducing kernel Hilbert space approach for Q-ball imaging”, IEEE Trans. Med. Imag. 30, 1877-1886.
Despite these FRT-based orientation, estimates can have lower resolution/accuracy relative to some of the alternative processing methods, see, Alexander, D. C., 2005, “Multiple-fiber reconstruction algorithms for diffusion MRI”, Ann. NY Acad. Sci. 1064, 113-133; Anderson, A. W., 2005. “Measurement of fiber orientation distributions using high angular resolution diffusion imaging”, Magn. Reson. Med. 54, 1194-1206; Jian, B., Vemuri, B. C., Özarslan, E., Carney, P. R., Mareci, T. H., 2007. “A novel tensor distribution model for the diffusion-weighted MR signal”, NeuroImage 37, 164-176; Ramirez-Manzanares, A., Rivera, M., Vemuri, B. C., Carney, P., Mareci, T., 2007, “Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry”, IEEE Trans. Med. Imag. 26, 1091-1102; Schultz, T., Seidel, H. P., 2008, “Estimating crossing fibers: A tensor decomposition approach”, IEEE Trans. Vis. Comput. Graphics 14, 1635-1646; Barmpoutis, A., Hwang, M. S., Howland, D., Forder, J. R., Vemuri, B. C., 2009, “Regularized positive-definite fourth order tensor field estimation from DWMRI”, NeuroImage 45, S153-S162; Descoteaux, M., Deriche, R., Knösche, T. R., Anwander, A., 2009, “Deterministic and probabilistic tractography based on complex fibre orientation distributions”, IEEE Trans. Med. Imag. 28, 269-286; Tournier, J. D., Yeh, C. H., Calamante, F., Cho, K. H., Connelly, A., Lin, C. P., 2008, “Resolving crossing fibres using constrained spherical deconvolution: Validation using diffusion-weighted imaging phantom data”, NeuroImage 42, 617-625; Rathi, Y., Michailovich, O., Shenton, M. E., Bouix, S., 2009. “Directional functions for orientation distribution estimation”, Med. Image Anal. 13, 432-444; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010, “Reconstruction of the orientation distribution function in single-and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009. “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650; Michailovich, O., Rathi, Y., 2010, “On approximation of orientation distributions by means of spherical ridgelets”, IEEE Trans. Image Process, 19, 461-477; Yeh, F. C., Wedeen, V. J., Tseng, W. Y. I., 2011, “Estimation of fiber orientation and spin density distribution by diffusion deconvolution”, NeuroImage 55, 1054-1062; Canales-Rodrìguez, E. J., Lin, C. P., Iturria-Medina, Y., Yeh, C. H., Cho, K. H., Melie-Garcìa, L., 2010. “Diffusion orientation transform revisited”, NeuroImage 49, 1326-1339; Yeh, F. C., Wedeen, V. J., Tseng, W. Y. I., 2010, “Generalized q-sampling imaging”, IEEE Trans. Med. Imag. 29, 1626-1635; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2010, “A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform”, NeuroImage 49, 1301-1315; Assemlal, H. E., Tschumperlè, D., Brun, L., Siddiqi, K., 2011, “Recent advances in diffusion MRI modeling: Angular and radial reconstruction”, Med. Image Anal. 15, 369-396, partly because the approach is based on a sub-optimal definition of the ODF, see Barnett, A., 2009, “Theory of q-ball imaging redux: Implications for fiber tracking”, Magn. Reson. Med. 62, 910-923; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010, “Reconstruction of the orientation distribution function in single- and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009, “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650.
Diffusion MRI and the Funk-Radon Transform
Diffusion MRI Basics
The statistical characteristics of molecular diffusion within a single MR imaging voxel may be coarsely summarized using a probability density function P(x,Δ) known as the ensemble average diffusion propagator (EAP), see Callaghan, P. T., 1991, “Principles of Nuclear Magnetic Resonance Microscopy” Clarendon Press, Oxford. The EAP quantifies the average probability that a water molecule, undergoing a random walk over a time period of length Δ, will be found at a spatial displacement of xε3 relative to its starting position. Under the assumptions of pure diffusion and that the system is at thermal equilibrium, the EAP may be symmetric (i.e., P(x,Δ)=P(−x, Δ)) and have zero-mean. Under the q-space formalism, see Callaghan, P. T., 1991, “Principles of Nuclear Magnetic Resonance Microscopy” Clarendon Press, Oxford, the ideal measured DW-MRI data may be modeled as (1), where the q-space function E(q) corresponds to the signal observed from the MR scanner as a function of diffusion gradient parameters q, and ƒ(x)∝P(x, Δ).
In practice, the diffusion signal E(q) may be measured over a finite set of different q values, and a variety of different q-space sampling schemes have been proposed. One approach is to densely sample q-space. For example, assuming the EAP is support-limited and the sampling density satisfies the Nyquist criterion, then reconstruction of the full EAP is possible via Fourier inversion, see Wedeen, V. J., Hagmann, P., Tseng, W. Y. I., Reese, T. G., Weisskoff, R. M., 2005. “Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging”, Magn. Reson. Med. 54, 1377-1386. However, this kind of encoding can be demanding on MR hardware, and can also be time consuming since experiment time is proportional to the number of samples. At the other extreme, parametric models for the EAP can be used to reduce q-space sampling requirements significantly. For example, the conventional diffusion tensor imaging (DTI) model, see Basser, P. J., Mattiello, J., LeBihan, D., 1994, “Estimation of the effective self-diffusion tensor from the NMR spin echo”, J. Magn. Reson. B 103, 247-254, is equivalent to modeling the EAP as a zero-mean Gaussian distribution, and estimation of the corresponding 3×3 positive semidefinite covariance matrix can be achieved with as few as 7 different q-space samples. However, a simple Gaussian model may be insufficient to accurately model diffusion in complex biological tissues. In the CNS, this may be due in part to the fact that the simple Gaussian model can have at most one dominant orientation, and thus may be incapable of representing the EAPs that are observed when a single voxel contains partial volume contributions from multiple crossing white matter fiber bundles.
HARDI sampling, in which diffusion data is sampled densely on the surface of a sphere, may enable a more reasonable balance between data acquisition speed/complexity and the capability to resolve crossing fibers. It has been shown that approaches using HARDI sampling can have only incrementally lower performance compared to Nyquist-rate Fourier encoding, but have significantly enhanced performance over DTI, see Kuo, L. W., Chen, J. H., Wedeen, V. J., Tseng, W. Y. I., 2008. “Optimization of diffusion spectrum imaging and q-ball imaging on clinical MRI system”, NeuroImage 41, 7-18; Zhan, W., Yang, Y., 2006, “How accurately can the diffusion profiles indicate multiple fiber orientations? a study on general fiber crossings in diffusion”, MRI. J. Magn. Reson. 183, 193-202.
Orientation Distribution Functions and the Funk-Radon Transform
An important objective in DW-MRI may be to derive fiber orientation information from HARDI data, and there may be different ways to define an ODF to quantify the orientations associated with an EAP. In QBI, see Tuch, D. S., 2004, “Q-ball imaging”, Magn. Reson. Med. 52, 1358-1372, an ideal ODF was defined as
                                                                                          ODF                  QBI                                ⁡                                  (                  u                  )                                            =                            ⁢                                                ∫                  0                  ∞                                ⁢                                                      f                    ⁡                                          (                                              α                        ⁢                                                                                                  ⁢                        u                                            )                                                        ⁢                                                                          ⁢                  d                  ⁢                                                                          ⁢                  α                                                                                                        =                            ⁢                                                1                  2                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                            f                      ⁡                                              (                                                  α                          ⁢                                                                                                          ⁢                          u                                                )                                                              ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    α                                                                                                                          =                            ⁢                                                1                  2                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                                                                              f                          ~                                                u                                            ⁡                                              (                                                  0                          ,                          0                          ,                          z                                                )                                                              ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    z                                                                                                          (        3        )            for orientation vectors uε12. The ODF definition from Tuch, D. S., 2004, “Q-ball imaging”, Magn. Reson. Med. 52, 1358-1372 included an additional dimensionless normalization constant, which is not included here to simplify notation.
The function {tilde over (ƒ)}u(•) expresses ƒ(•) in a rotated coordinate system in which the orientation vector u is parallel to the z-axis. This definition of the ODF computes radial projections of the EAP, which Tuch showed can be approximated by the FRT. In particular, it can be verified that the FRT satisfies
                                                                                        ⁢                                  E                  ⁡                                      (                    q                    )                                                  ⁢                                  δ                  ⁡                                      (                                                                  u                        T                                            ⁢                      q                                        )                                                  ⁢                d                ⁢                                                                  ⁢                q                            =                            ⁢                              ρ                ⁢                                                      ∫                    0                                          2                      ⁢                      π                                                        ⁢                                                                                                              E                          ~                                                u                                            ⁡                                              (                                                  ρcosθ                          ,                          ρsinθ                          ,                          0                                                )                                                              ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    θ                                                                                                                          =                            ⁢                              2                ⁢                πρ                ⁢                                  ∫                                      ∫                                                                ⁢                                                                                                    f                            ~                                                    u                                                ⁡                                                  (                                                      x                            ,                            y                            ,                            z                                                    )                                                                    ⁢                                                                        J                          0                                                ⁡                                                  (                                                      2                            ⁢                            πρ                            ⁢                                                                                                                            x                                  2                                                                +                                                                  y                                  2                                                                                                                                              )                                                                    ⁢                      d                      ⁢                                                                                          ⁢                      x                      ⁢                                                                                          ⁢                      d                      ⁢                                                                                          ⁢                      y                      ⁢                                                                                          ⁢                      d                      ⁢                                                                                          ⁢                      z                                                                                                                              (        4        )            where {tilde over (E)}u (•) is the coordinate-rotated version of E(•) (cf. {tilde over (ƒ)}u(•) and ƒ(•)), and J0(•) is the zeroth-order Bessel function of the first kind. Given that J0(•) has most of its energy concentrated at the origin, (4) can be viewed as a coarse approximation of (3), with the approximation quality improving for larger values of the sampling radius ρ.
While the QBI definition of the ODF has been used, an improved ODF definition may be given by Wedeen, V. J., Hagmann, P., Tseng, W. Y. I., Reese, T. G., Weisskoff, R. M., 2005, “Mapping complex tissue architecture with diffusion spectrum magnetic resonance Imaging”, Magn. Reson. Med. 54, 1377-1386; Barnett, A., 2009, “Theory of q-ball imaging redux: Implications for fiber tracking”, Magn. Reson. Med. 62, 910-923; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010, “Reconstruction of the orientation distribution function in single- and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009. “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650
                                                                        ODF                ⁡                                  (                  u                  )                                            =                            ⁢                                                ∫                  0                  ∞                                ⁢                                                      f                    ⁡                                          (                                              α                        ⁢                                                                                                  ⁢                        u                                            )                                                        ⁢                                      α                    2                                    ⁢                                                                          ⁢                  d                  ⁢                                                                          ⁢                  α                                                                                                        =                            ⁢                                                1                  2                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                            f                      ⁡                                              (                                                  α                          ⁢                                                                                                          ⁢                          u                                                )                                                              ⁢                                          α                      2                                        ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    α                                                                                                                          =                            ⁢                                                1                  2                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                                                                              f                          ~                                                u                                            ⁡                                              (                                                  0                          ,                          0                          ,                          z                                                )                                                              ⁢                                          z                      2                                        ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                                          z                      .                                                                                                                              (        5        )            
Physically, this may correspond to radial integration of the EAP over a cone of constant solid angle, and differs from (3) by including the appropriate volume element for radial integration in spherical coordinates. And unlike (3), this ODF may be proportional to the probability distribution that would be obtained by marginalizing over the radial component of the EAP. While (5) may have better theoretical and practical characteristics than the QBI ODF, see Barnett, A., 2009, “Theory of q-ball imaging redux: Implications for fiber tracking”, Magn. Reson. Med. 62, 910-923; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010, “Reconstruction of the orientation distribution function in single-and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009. “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650; Assemlal, H. E., Tschumperlè, D., Brun, L., Siddiqi, K., 2011, “Recent advances in diffusion MRI modeling: Angular and radial reconstruction”, Med. Image Anal. 15, 369-396, existing methods to compute (5) from sampled data may have either required additional modeling assumptions, see; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009. “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010, “Reconstruction of the orientation distribution function in single- and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2010, “A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform”, NeuroImage 49, 1301-1315; Canales-Rodrìguez, E. J., Lin, C. P., Iturria-Medina, Y., Yeh, C. H., Cho, K. H., Melie-Garcìa, L., 2010, “Diffusion orientation transform revisited”, NeuroImage 49, 1326-1339; Assemlal, H. E., Tschumperlè, D., Brun, L., Siddiqi, K., 2011. “Recent advances in diffusion MRI modeling: Angular and radial reconstruction”, Med. Image Anal. 15, 369-396, the application of nonlinear operators (e.g., logarithmic-transformations) to the diffusion data, see Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2009. “Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging”, NeuroImage 47, 638-650; Aganj, I., Lenglet, C., Sapiro, G., Yacoub, E., Ugurbil, K., Harel, N., 2010, “Reconstruction of the orientation distribution function in single- and multipleshell q-ball imaging within constant solid angle”, Magn. Reson. Med. 64, 554-566; Tristàn-Vega, A., Westin, C. F., Aja-Fernàndez, S., 2010, “A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform”, NeuroImage 49, 1301-1315; Canales-Rodrìguez, E. J., Lin, C. P., Iturria-Medina, Y., Yeh, C. H., Cho, K. H., Melie-Garcìa, L., 2010, “Diffusion orientation transform revisited”, NeuroImage 49, 1326-1339, or more complicated non-spherical q-space sampling, see Wedeen, V. J., Hagmann, P., Tseng, W. Y. I., Reese, T. G., Weisskoff, R. M., 2005, “Mapping complex tissue architecture with diffusion spectrum magnetic resonance Imaging”, Magn. Reson. Med. 54, 1377-1386; Canales-Rodrìguez, E. J., Melie-Garcìa, L., Iturria-Medina, Y., 2009, “Mathematical description of q-space in spherical coordinates: Exact q-ball imaging”, Magn. Reson. Med. 61, 1350-1367.